If it ever existed, what David Cameron calls "state multiculturalism" died 10 years ago this month, on the streets of Oldham. Following months of simmering tension and violence, the Oldham disturbances were triggered by attacks on the edge of an Asian district and escalated into battles between (largely) young Asian men and police. A month later, similar riots broke out in Burnley and then – in July – Bradford saw the most serious riots of all.

Local and national investigations into the riots concurred that a principal cause was a deep division between white and Asian communities, exacerbated by local authority policies, from supposed bans on flying English flags to multilingual leaflets, which the white community perceived as privileging Asians. Ted Cantle's report to the Home Office recommended a new policy of "community cohesion", advocating a concept of citizenship based on "a clear primary loyalty to this nation".

Cantle's recommendations were reiterated in the aftermath of the 2005 tube and bus bombings. In the interim the home secretary, David Blunkett, had introduced legislation for tests that would make new citizens more au fait than most of us with electricity voltages, how to get a national insurance number and the powers of the Welsh assembly. Shortly after 7/7, Trevor Phillips of the Commission for Racial Equality made his "sleepwalking into segregation" speech. The following January Gordon Brown suggested to a Fabian conference that British households should follow the American practice of planting flags in their lawns.

That December Tony Blair summed up what was clearly the new political consensus: "Our tolerance is part of what makes Britain Britain. So conform to it; or don't come here."

So why – 10 years from what was in effect its demise – does "state multiculturalism" remain such a bogeyman? Why indeed did Cameron choose to savage this ex-policy, in his Munich speech in February, as if he were the first person to do so? Why did he go on to propose a to-do list – from making knowledge of English compulsory for new citizens to cold-shouldering "non-violent extremists" – that had been the Labour government's policy for years?

There is an international context to this. David Cameron's February speech was delivered in Germany, against the background of a chorus of speeches by European leaders rejecting multiculturalism not just as a government policy but as a fact, on the basis not of liberal, enlightenment values but those of Christian tradition. Increasingly, the European centre-right has moved beyond withdrawing of state encouragement for minority cultures, towards advocating or imposing thoroughly illiberal restrictions on minority religious rights. Will British conservatives follow their continental colleagues in banning veils and minarets, forbidding civil servants from wearing headscarves, removing halal food from schools and discouraging foreign languages in public places?

Or was Cameron just playing an old political game? This is the view of the ConservativeHome blogger Tim Montgomerie in his tweet: "Increasingly nervous about core Tory vote, Cameron makes immigration speech". In his second speech on race this year – in Hampshire last month – Cameron spoke warmly of Britain's "real communities", knitted together by common experiences and rituals, "from the school run to the chat down the pub". This cosy communitarian image excludes – of course – those who don't visit pubs or drink warm beer, but it's there to contrast with the following paragraph, in which Cameron speaks ominously of "new people" arriving, "perhaps not able to speak the same language", sometimes "not really wanting or even willing to integrate", creating "a kind of discomfort and disjointedness in some neighbourhoods".

By lauding idealised notions of community, Cameron may be appealing not just to his own core vote but to that section of Labour's that is being wooed by the new, "Blue Labour" tendency in the party. Maurice Glasman's rejection of the neoliberal allure of globalisation, in favour of the communal virtues of locality and solidarity, is attractive. But, as the recent Radio 4 feature by the former Prospect editor David Goodhart demonstrated, left communitarianism can easily mutate into a nativism in which solidarity across ethnic divisions can be dismissed as yet another fad imposed by metropolitan liberals on once homogeneous working-class communities. Once again, Labour is being tempted to outflank the Tories to the right.

In fact, Britain's working-class communities have been redefined and reinvigorated by newcomers for centuries. Industrial struggles led by black and Asian workers now have a 40-year history. While researching a play about the 2001 riots for the National Theatre, I was struck by how Oldham and Burnley's Asian districts felt and looked like the family-based working-class communities of the 1950s; indeed, in a way they stood as a reproach to white estates that had lost that sense of identity, cohesion and pride.

In considering what might follow the straw man of "state multiculturalism", it's worth looking back at Oldham, Burnley and Bradford and remembering that the riots began as a community response to the rumour or the fact of far-right attacks on Asian neighbourhoods. Recently, the anti-fascist magazine Searchlight identified large swaths of the population vulnerable to far-right propaganda. However, its activists on the ground have ably demonstrated how, if confronted at community level, the far right can be annihilated electorally – as, in all but two council seats, the BNP was last Thursday. In the 2001 general election Nick Griffin gained 16% of the vote in Oldham West. Ten years on there were no BNP candidates in Oldham at all. This is the result not of an imposed multi- or monoculturalism, but disparate and diverse elements of living communities coming together against a common enemy.

• A radio version of David Edgar's play about the 2001 riots - Playing with Fire - is being broadcast by Radio 4 on 23 July